Chapter 1 Testing Gravity with Black Hole X-Ray Data Cosimo Bambi_2

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Chapter 1

Testing Gravity with Black Hole X-Ray Data

Cosimo Bambi

Abstract The analysis of the properties of the X-ray radiation emitted from geo-

metrically thin accretion disks around black holes can be a powerful tool to test

General Relativity in the strong ﬁeld regime. This chapter reviews the state-of-the-

art of gravity tests with black hole X-ray data. So far, most efforts have been devoted

to test the Kerr hypothesis – namely that the spacetime around astrophysical black

holes is described by the Kerr solution – and X-ray data can currently provide among

the most stringent constraints on possible deviations from the Kerr geometry. As of

now, all X-ray analyses are consistent with the predictions of General Relativity.

1.1 Introduction

In 4-dimensional General Relativity, black holes are completely speciﬁed by three

parameters: the black hole mass, the black hole spin angular momentum, and the

black hole electric charge. This is the result of the celebrated no-hair theorem, which

is actually a family of theorems with speciﬁc assumptions and a number of exten-

sions [36,105,38]. Uncharged black holes are described by the Kerr solution [70]

and have only two parameters: the black hole mass and the black hole spin angular

momentum.

The spacetime around astrophysical black holes formed by the complete grav-

itational collapse of a progenitor astrophysical body should be described well by

the Kerr solution. For example, initial deviations from the Kerr metric are quickly

radiated away by the emission of gravitational waves at the time of the forma-

tion of the black hole [95]. The presence of nearby stars or of an accretion disk

have normally a negligible impact on the spacetime geometry near the black hole

Cosimo Bambi ( )

Center for Field Theory and Particle Physics and Department of Physics

Fudan University, 200438 Shanghai, China

e-mail: bambi@fudan.edu.cn

arXiv:2210.05322v1 [gr-qc] 11 Oct 2022

2 Cosimo Bambi

event horizon [19,27]. The black hole equilibrium electric charge resulting from

the difference between the proton and electron masses is tiny and can be ignored

for macroscopic objects [19,26]. The impact of these and other effects on the

spacetime metric around a black hole can be quantiﬁed, but it turns out that the

induced deviations from the Kerr solution are normally extremely small and negli-

gible even in the case of future very accurate tests of the Kerr hypothesis. On the

contrary, macroscopic deviations from the Kerr solution are predicted by a num-

ber of scenarios with new physics, from models with macroscopic quantum gravity

effects (see, e.g., Refs. [46,53]) to scenarios with exotic matter ﬁelds (see, e.g.,

Refs. [59,60]) or in the case General Relativity is not the correct theory of gravity

(see, e.g., Refs. [136,71]).

From astronomical observations, we know three classes of astrophysical black

holes: stellar-mass black holes, supermassive black holes, and intermediate-mass

black holes [20].

•Stellar-mass black holes are the natural product of the evolution of very heavy

stars. When a star exhausts all its nuclear fuel, the thermal pressure of the plasma

cannot compensate the gravitational force any longer and the body shrinks to

ﬁnd a new equilibrium conﬁguration. If the quantum pressure of electrons or

neutrons can compensate the weight of the collapsing part of the star, we have

the formation of, respectively, a white dwarf or a neutron star. If the collapsing

body is too heavy, there is no mechanism to stop the collapse and we have the

formation of a black hole.

The minimum mass of these black holes is thus set by the Oppenheimer-Volkof

limit, which is the maximum mass for a neutron star and is around 2-3 M,

depending on the exact matter equation of state, composition, rotation, etc. [76].

The maximum mass for stellar-mass black holes is probably around 100 Mfor

objects formed by the direct collapse of primordial metal-poor stars of about

100 M[84]. For heavier stars, the gravitational collapse is so violent that may

destroy the whole system, without leaving any remnant. For stars with higher

metallicity, the outer envelope of the star is ejected into space (heavier elements

have larger photon cross-sections) and the mass of the ﬁnal black hole cannot be

higher than 20-30 M[84].

From stellar evolution studies, we expect a population of 108-109stellar-mass

black holes in a galaxy like the Milky Way [116]. While this is a huge num-

ber, it is extremely difﬁcult to identify these objects and, as a result, the number

of known stellar-mass black holes is much lower. From electromagnetic obser-

vations, we currently know about 70 stellar-mass black holes in X-ray binary

systems, and only for about 25 objects we have a dynamical measurement of

the mass (i.e., from the study of the orbital motion of the companion star we

can infer that the mass of the black hole exceeds the Oppenheimer-Volkof limit

and therefore it cannot be a neutron star); see Fig. 1.1. The majority of these

∼70 stellar-mass black holes are in the Milky Way, while only a few of them are

in nearby galaxies.

Most of the known black hole binaries are transient X-ray sources: they are nor-

mally in a quiescent state with a very low X-ray luminosity (they can be even

1 Testing Gravity with Black Hole X-Ray Data 3

Fig. 1.1: Sketch of 22 X-ray binaries with a stellar-mass black hole conﬁrmed by

dynamical measurements. For every system, the black hole accretion disk is on the

left and the companion star is on the right. The color of the companion star roughly

indicates its surface temperature (from brown to white as the temperature increases).

We can compare the size of these X-ray binaries with the system Sun-Mercury in

the top left corner: the distance Sun-Mercury is about 50 million km and the radius

of the Sun is about 0.7 million km. Figure courtesy of Jerome Orosz.

too faint to be detected by our X-ray observatories) and sometimes they have

an outburst, when there is a signiﬁcant transfer of material from the companion

star to the black hole. Every year, we may discover 1-3 new black holes, when

their binary system has an outburst (see Fig. 1.2). On the other hand, persistent

4 Cosimo Bambi

Fig. 1.2: Cumulative histogram of the number of discovered stellar-mass black holes

in transient X-ray sources. Red bars are for new black holes and blue bars are for

dynamically conﬁrmed black holes. The horizontal gray bars show the main X-ray

missions used to discover and study these black holes. From the online BlackCAT

catalog https://www.astro.puc.cl/BlackCAT/ of Ref. [39].

sources are relatively rare: in Fig. 1.1, only Cygnus X-1, LMC X-1, LMC X-3,

and M33 X-7 are persistent X-ray sources and all other X-ray binaries are tran-

sients. GRS 1915+105 is quite a peculiar case: it started its outburst in 1992 and

since then it appears as a persistent X-ray source in the sky.

From gravitational wave observations, so far we have detected about 90 events

in which two stellar-mass black holes (or a stellar-mass black hole and a neutron

star or two neutron stars) merged to form a heavier black hole; see Fig. 1.3. With

the current sensitivity of the LIGO and Virgo experiments, we can detect a new

merger event every few days, but the detection rate will signiﬁcantly increase

with the next generation of gravitational wave observatories.

•Supermassive black holes are black holes with a mass in the range 105-1010 M

and are found in galactic nuclei. Every middle-size or large galaxy seems to have

a supermassive black hole at its center, while in the case of small galaxies the

situation is more controversial: some small galaxies may have a supermassive

black hole but other small galaxies may not.

While heavier objects can naturally migrate to the center of a multi-body system,

and therefore it is not a surprise to ﬁnd such supermassive objects in galactic

nuclei, their exact origin is not completely understood: it is not like the case of

1 Testing Gravity with Black Hole X-Ray Data 5

Fig. 1.3: Stellar-mass black holes and neutron stars with a robust mass measurement.

Black holes (neutron stars) discovered with gravitational waves are in blue (orange)

and black holes (neutron stars) in X-ray binaries are in magenta (green). Credit:

LIGO-Virgo-KAGRA/Aaron Geller/Northwestern

stellar-mass black holes, which should be anyway expected as the ﬁnal product

of the evolution of very heavy starts. Supermassive black holes were certainly

much smaller when they formed and have grown from merger with other black

holes and accretion of the surrounding material. However, it is puzzling to ob-

serve black holes with masses of about 1010 Min high-redshift galaxies, when

the Universe was only 1 Gyr old; see, e.g., Ref. [132]: a stellar-mass black hole

formed from the ﬁrst generation of stars would not have had the time to grow

so much in such a short time without exceeding the Eddington limit. It is pos-

sible that the supermassive black holes we see today in galactic nuclei formed

from the direct collapse of heavy clouds, and therefore their initial mass was a

few order of magnitudes higher than the maximum mass of stellar-mass black

holes [128]. They may have also experienced some period of super-Eddington

accretion and/or grown from the merger of several black holes [128]. The ac-

tual mechanism is currently unknown, but it will be investigated by the next

generation of gravitational wave observatories, which will have the capability of

detecting black hole mergers at very high redshift, potentially even before the

formation of the ﬁrst stars.

•Intermediate-mass black holes are black holes with a mass ﬁlling the gap be-

tween the stellar-mass and the supermassive black holes. However, unlike in the

case of stellar-mass and supermassive black holes, so far there are no robust mea-

surements of the masses of these objects, so technically we should speak about

intermediate-mass black hole “candidates”.

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Chapter1TestingGravitywithBlackHoleX-RayDataCosimoBambiAbstractTheanalysisofthepropertiesoftheX-rayradiationemittedfromgeo-metricallythinaccretiondisksaroundblackholescanbeapowerfultooltotestGeneralRelativityinthestrongeldregime.Thischapterreviewsthestate-of-the-artofgravitytestswithblackholeX-rayd...

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